Vectors

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Vectors
It doesn’t tell
you where to start.
E.g. you are told to
move five seats to
the left.
A vector is an instruction that tells you how far to go
and which direction to go in.

Vectors and scalars
Examples of vector quantities are:
Examples of scalar quantities are:
velocity
displacement
force
speed
length
mass
A scalar is a quantity that has size (or magnitude) only.
A vector is a quantity that has both size (or magnitude)
and direction.

Representing vectors
A vector can be represented using a line segment with an
arrow on it.
For example,
A
B
The magnitude of the vector is given by the length of the line.
The direction of the vector is given by the arrow on the line.

A
B
This vector goes from the point A to the point B.
We can write this vector as AB.
Vectors can also be written using single letters in bold type.
For example, we can call this vector a.
a
When this is hand-written, the a is written as a

A
B
This vector tells you to go six chairs to the right,
and three chairs up.
It doesn’t matter which seat you are sat in to start with.

A
B
We can represent this movement using a column vector.
AB =
6
3
This is the horizontal component. It tells
us the number of units in the x-direction.
This is the vertical component. It tells us
the number of units in the y-direction.
6
3

Equal vectors
Two vectors are equal if they have the same magnitude
and direction.
All of the following vectors are equal:
They are the same length and parallel to each other.
a
b
c
d
e
f
g
They are the same length and parallel to each other.

The negative of a vector
Here is the vector AB =
5
2
a
A
B
Suppose the arrow went
in the opposite direction:
A
B
We can describe this vector as:
BA –a
–5
–2
or
How can we describe this vector?

a
A
B
–a
A
B
If this is the vector a, this is the vector –a.
The negative of a vector is the same length and has the
same slope, but goes in the opposite direction.
In general,
if a =
x
y
then –a =
–x
–y

A negative vector says that if one bottle moves five left and
two up, then another bottle moves five right and two down.

The zero and unit vector
A vector with a magnitude of 0 is called the zero vector.
The zero vector is written as 0 or hand-written as 0
A vector with a magnitude of 1 is called a unit vector.
The most important unit vectors are those that are horizontal
and vertical. These are called unit base vectors.
The horizontal unit base vector, , is called i.
1
0
The vertical unit base vector, , is called j.
0
1

The unit base vectors
The unit base vectors, i and j, can be represented in a
diagram as follows:
j
i
Any column vector can easily be written in terms of i and j.
For example,
5
–4
= 5i – 4j
The number of i’s tells us how many units are moved
horizontally and the number of j’s tell us how many units are
moved vertically.

The unit base vectors
7i + 2j
7
2
=
i – 3j
1
–3
=
–5i
–5
0
=
Write the following in
terms of unit base vectors.
Write the following in
terms of column vectors.
4
1
4i + j =
0
–7
–7j =
–1
8
–i + 8j =

Multiplying vectors by scalars
Remember, a scalar quantity has size but not direction.
A scalar quantity can be represented by a single number.
A vector can be multiplied by a scalar.
The vector 2a has the same
direction but is twice as long.
a =
3
2
2a =
6
4
a 2a

In general, if the vector is multiplied by the scalar k, thenx
y
x
y
k × =
kx
ky
When a vector is multiplied by a scalar the resulting vector is
either parallel to the original vector or lies on the same line.
For example,
–2
5
3 × =
–6
15
Can you explain why this is?

Pairs – parallel vectors

Adding vectors
Adding vectors is like
saying move 2 left
and 6 up then move
5 right and 1 down.
From your starting position you
have moved 3 right and 5 up.
3
5
How would you
describe this
more simply?

Adding vectors
Adding two vectors is equivalent to applying one vector
followed by the other. For example,
Suppose a =
5
3
and b =
3
–2
Find a + b
We can represent this addition in the following diagram:
a b
a + b
a + b =
8
1

Adding vectors
When two or more vectors are added together the result is
called the resultant vector.
In general, if a =
a
b
and b =
c
d
Add two column vectors by adding the horizontal components
together and adding the vertical components together.
a + b =
a + c
b + d

Adding vectors

Adding vectors
Adding vectors is used in real life when you are calculating
more than one force operating on something.
Sailors have to
consider both
water currents
and wind, when
plotting direction.

Adding vectors
Meteorologists use vectors to map out weather patterns.
Wind speed can be mapped by vectors indicating direction
of the wind with their length indicating intensity.

Subtracting vectors
We can subtract two column vectors by subtracting the
horizontal components and subtracting the vertical
components. For example,
Find a – b
Suppose and b =
–2
3
a =
4
4
a – b =
4
4
–
–2
3
=
4 – –2
4 – 3
=
6
1

Subtracting vectors
To show this subtraction in a diagram, we can think of a – b
as a + (–b).
and b =
–2
3
a =
4
4
a
b
a – b
a – b =
6
1
–b a
–b

Adding and subtracting vectors

The resultant vector

The parallelogram law for vector addition
As you have seen, we can use a parallelogram to
demonstrate the addition of two vectors.
and b =
2
3
a =
4
–1
a
a + b
b
B
C
From this diagram we can see
that
a + bAC = AB + BC =
a
b
Also
b + aAC = AD + DC =
a
Vector addition is commutative
A
D

Vectors

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